Generative Structural Theory v1.101 — A Depth-Two Probing Counterexample with No Additional Core Deletion

For the one-sided re-hangings of a transit leaf studied since v1.91, with D = ∅ (no pair of the core deleted in addition to the re-hanging), v1.98 asked whether the probing rule applied at depth two decides "impossible" at every undecided stop without a feasible set ((Q32)); v1.99 defined the rule of every depth and proved it sound, and v1.99 and v1.100 gave sufficient conditions for depth two. This text answers (Q32) in the negative. Lemma ϱ1: at a node passed through once (exactly one pair into it and one out of it among the pairs of the core, the extra pair v and the usable pairs, and both of these pairs of the core), neither pair can be bypassed. Proposition ϱ2: at the input J13a on thirteen nodes, the input of the stop A12-04 of v1.100 with one pair of the relation divided by a new node, the forcing procedure stops undecided at once, no feasible set exists, and the rule of depth two does not decide "impossible". Its proof is a symbolic argument on this finite graph, supported by explicitly listed reachability and strong-connectivity checks. Five cuts and ten obstructions exclude every feasible set. To establish failure of depth-two probing, five candidate pairs meet every cut's candidate set; forcing any one of these five pairs leads to an undecided stop at which depth-one probing also fails to decide "impossible". So depth two does not detect every infeasibility, even with no additional core deletion (Corollary ϱ3); if some depth decides every such stop, it is at least three. Findings of finite cases (no ground for a general statement): after a preliminary exploration, disclosed, search plan 02 was fixed and digested before the main check, on the one-node extensions to thirteen nodes of the inputs of the four stops with k = 4 of v1.100; it lists 112 stops missed at depth two, counted over the four families, all made by dividing one pair of the relation, their inputs in 93 classes under the isomorphism named, each confirmed by a second implementation and decided "impossible" at depth three. J13a is a counterexample on thirteen nodes, not known to be least. (Q33), (Q34), the new question (Q35) on a missed stop with fewer nodes, the rest of (Q27) of v1.93, (Q28) of v1.93, (Q29) of v1.95, (Q22) of v1.88 and (Q1) of v1.76 are open; the result settles one limit of the local rule studied for the two-deletion chains of (Q1), and does not settle (Q1).